- Email: [email protected]
Phase space quantization and the uncertainty principle
Physics Letters A 317 (2003) 365–369 www.elsevier.com/locate/pla
Phase space quantization and the uncertainty principle Maurice A. de Gosson a,b a Blekinge Institute of Technology, SE-371 79 Karlskrona, Sweden 1 b University of Colorado at Boulder, Boulder, CO 80302, USA
Received 10 June 2003; received in revised form 31 August 2003; accepted 7 September 2003 Communicated by J.P. Vigier
Abstract We replace the usual heuristic notion of quantum cell by that of ‘quantum blob’, which does not depend on the dimension of phase space. Quantum blobs, which are defined in terms of symplectic capacities, are canonical invariants. They allow us to prove an exact uncertainty principle for semiclassically quantized Hamiltonian systems. 2003 Elsevier B.V. All rights reserved. PACS: 03.65.Ta; 03.65.Sq; 02.40 Keywords: Quantum cells; Quantum uncertainty; Symplectic non-squeezing; Lagrangian tori
In [4] Hall and Reginatto remark that ‘. . . uncertainty relations expressed as imprecise inequalities are not enough to pin down what the essence of what is not classical in quantum mechanics’. A similar statement certainly applies to the ‘coarse graining’ of phase space into ‘quantum cells’ familiar from semiclassical and statistical mechanics. To paraphrase Hall and Reginatto, imprecise notions of quantum cells are not enough to pin down the essence of phase space quantization! In fact the usual ‘definitions’ of such cells are unprofessionally vague and go no further than Heisenberg’s 1932 discussion of the microscope; they are a heuristic device from the beginning years of quantum mechanics, and have no fundamental meaning. For instance, a very common
E-mail address: [email protected] (M.A. de Gosson). 1 Current address.
0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2003.09.008
choice is to use cubic boxes with volume hn (n the number of degrees of freedom) and whose faces lie in the conjugate planes qj , pj . This choice is thought to be consistent with Heisenberg’s uncertainty principle written ‘approximately’ as pj qj h. What about spherical cells with same volume hn ? Since the volume of a ball with radius R in phase space R2n is π n R 2n /n!, Stirling’s formula shows that for large n the area of the projection of the sphere on any plane is πR 2 ∝ πnh; this number goes to infinity with the number of degrees of freedom of the system, while the area of the projection on a conjugate plane of a cubic cells remains equal to h. Last, but not least, any definition of quantum cells directly based on the Heisenberg relations is doomed to be badly dependent on the choice of coordinates. This is because the inequalities pj qj h are not canonically invariant, except for permutations pj ↔ qj or rescalings: pj → λpj , qj → pj /λ. If, for
366
M.A. de Gosson / Physics Letters A 317 (2003) 365–369
instance, we rotate the pj , qj axes by an angle of π/4 these inequalities become pi 2 − qi 2 h in the new coordinates. What should a decent definition of a quantum cell be in this case? Our purpose is to propose a precise definition of the notion of quantum cell avoiding these drawbacks. Our definition will not depend on the number of degrees of freedom; it is hence an intensive quantity, as opposed to the usual definitions in terms of volume. The definition of these cells—which we prefer to call ‘quantum blobs’—is made possible by a careful use of two results from symplectic topology, the non-squeezing theorems of Gromov and Sikorav. The first has been used in our previous work [1] where we showed that the ground energy level for completely integrable systems can be recovered by making a simple topological assumption involving Planck’s constant. Other possible applications and extensions are discussed at the end of this Letter. Two non-squeezing results Canonical transformations are not only volume preserving mappings, but they enjoy in addition a very special property. This property, called the ‘non-squeezing theorem’, alias the ‘principle of the symplectic camel’, was proven by Gromov [3] in 1985. His theorem says that if a phase space region R (in any number of degrees of freedom) contains a ball with given radius R, then if we deform R using canonical transformations only, the area of the projection of R on any of the conjugate planes qj , pj will never decrease beyond its original value πR 2 (whereas the areas of projections on non-conjugate planes such as q1 , p2 , q1 , q2 , etc. can become arbitrarily small). Gromov’s theorem was the first of a whole constellation of new results in symplectic topology (we will describe another non-squeezing result below). It is equivalent to the existence of symplectic capacities. These are mappings associating to every subset B of phase space a number 0, or +∞, and having the following properties: (I) c is a canonical invariant: c(f (B)) = c(B) for every canonical transformation f ; (II) c is monotone: c(B) c(B ) if B ⊂ B ; (III) c is 2homogeneous: c(λB) = λ2 c(λB) for every real λ; and, finally, (IV): c B(R) = c Zj (R) = πR 2 ,
where B(R) is a phase space ball with radius R and Zj (R) any phase space cylinder with same radius and based on the qj , pj plane. The properties (I–IV) are strongly reminiscent of those of the usual area of a plane surface. One proves that in fact the only symplectic capacity in the phase plane (n = 1) is precisely area. It turns out that Gromov’s theorem is equivalent to the existence of two distinguished symplectic capacities cG and cG (the ‘lower’ and ‘upper Gromov capacities’) such that cG (B) c(B) cG (B)
(1)
for all B ⊂ R2n and every symplectic capacity c. The Gromov capacities are defined as follows: cG (B) = πR 2 where R is the supremum of the radii of all balls that can be sent in B using canonical transformations; cG (B) is the infimum of the radii of all cylinders Zj (R) into which B can be sent using canonical transformations. Another essential result is that of Sikorav [10], which is stated and proved in detail in Polterovich [9, Theorem 1.1C, p. 2, and §4.3]. Sikorav’s result is about Lagrangian tori; it says the following: let Sj1 (R) be the circle qj2 + pj2 = R 2 lying in the qj , pj plane, and T(R) = S11 (R) × · · · × Sn1 (R) the Lagrangian torus obtained by taking the Cartesian product of all these circles. Then there exists a canonical transformation sending T(R) in the cylinders Zj (r) if and only if R r; in other words: cG T(R) = πR 2 . (2) In view of the definition of the upper Gromov capacity cG Sikorav’s theorem means that the infimum of the radii of all cylinders Zj into which B can be sent using canonical transformations is equal to R, the radius of each factor Sj1 (R) of the torus T(R). Quantum blobs Let B be a subset of R2n . It follows from Gromov’s theorem and the inequalities (1) that if there exists a canonical transformation F such that B(R) ⊂ F (B) ⊂ Zj (R) then c(B) = πR 2 for every symplectic capacity c. We will call such sets B admissible. The image of an admissible set by a canonical transformation is of course still admissible.
M.A. de Gosson / Physics Letters A 317 (2003) 365–369
We will call quantum blob any admissible subset Bquant of R2n such that c(Bquant) = 12 h. In view of Gromov’s theorem the projection of Bquant on any of the conjugate coordinate planes qj , pj encloses a surface of area at least 12 h. However, to the difference of any traditional quantum cell, a quantum blob can be unbounded, and √even have infinite volume: any of the cylinders Zj ( h¯ ) is a quantum blob in its own right. Another difference is that while the volume of a cell depends on the number of particles, that is on the dimension of phase space, quantum blobs are defined in terms of the quantum of action which has the dimension of an area. We also notice that the notion of quantum blob is a canonical invariant, that is c(f (B)) = c(B) for every canonical transformation f : R2n → R2n (this follows from the very definition of a symplectic capacity). From now on we assume that H = H (q, p) is a completely integrable Hamiltonian. The semiclassical motion takes place on Lagrangian manifolds (‘invariant tori’) V which we assume quantized by the usual EBK condition 1 1 p dq − m[γ ]h¯ is an integer 0, 2π 4 γ
for every one-cycle γ is on the manifold V and m[γ ] its Maslov index (see Maslov [6]; Maslov and Fedoriuk [7]; also de Gosson [2]). We will call such motions ‘quantized motions’. Let us focus in particular on the ground state. The EBK condition is in this case 1 1 (3) p dq = m[γ ]h¯ . 2π 4 γ
We will denote a Lagrangian manifold satisfying this condition by Vground . We claim that: (A). Any motion on Vground takes places on a quantum blob (in [1] we proved the converse of this property). Proof. Passing to action-angle variables (I, φ) and defining then new canonical variables by Qj = 2Ij × cos φj and Pj = 2Ij sin φj the motion takes place on tori T = S1 2I1 (0) × · · · × Sn 2In (0) . Now, the EBK condition (3) for the ground level imposes that Ij (0) = 12 h¯ (because the Maslov index is
367
equal to 2 along circles), hence the ground-level torus is just √ √ Tground = S1 h¯ × · · · × Sn h¯ . In particular, the trajectory winds around each cylindrical quantum blob √ Zj h¯ : Q2j + Pj2 = h¯ . (4) Returning to the original coordinates q, p amounts to use successively the canonical transformations (Q, P ) → (I, φ) and (I, φ) → (q, p); this proves completely the statement since quantum blobs are canonical invariants. ✷ We will see below (property (C)) that the result (A) is actually sharp, in the sense that a quantized motion cannot be carried by an admissible set with symplectic capacity < 12 h. Quantum uncertainty We are going to precise statement (A). Let us call ‘simple surface’ in a plane a surface whose boundary can be smoothly deformed into a circle. We contend that following topological version of the Heisenberg uncertainty principle holds: (B). Any simple surface in a qj , pj plane containing the projection of a quantized motion has area at least 12 h; this condition remains true in all canonical coordinates. Proof. We begin by noting that in view of (2) we have cG (Vground) = 12 h. Let now λj be a simple closed curve in the qj , pj plane such that the surface Λj it bounds contains the projection of Vground . Let us first assume that λj is a circle Sj1 (R). Then Vground ⊂ Zj (R) and hence 1 h = cG (Vground) cG Zj (R) = πR 2 2 proving the claim in that case. In the general case choose a diffeomorphism f of the qj , pj plane taking Λj into a circle Sj1 (R) with same area; in view of a classical result of Moser and Dacorogna [8] we may assume that f is area-preserving. Let F be the phase space mapping transforming (qj , pj ) into f (qj , pj ) and leaving all other coordinates unchanged; that mapping is canonical, and the projection of F (Vground)
368
M.A. de Gosson / Physics Letters A 317 (2003) 365–369
lies inside Sj1 (R). Now cG (Vground) = cG F (Vground) cG Zj (R) = Area(Λj )
1 2 h, and this in any number of degrees of freedom. Can
we use them to count quantum states in the usual way? Consider Weyl’s rule (5)
hence again Area(Λj ) 12 h. That the result holds in all canonical coordinates follows from the invariance of the EBK conditions under canonical transformations. ✷ What about the excited states of an integrable Hamiltonian system? Sikorav’s theorem obviously implies that cG S11 (R1 ) × · · · × Sn1 (Rn ) π inf Rj2 . 1j n
It follows that if the torus S11 (R1 ) × · · · × Sn1 (Rn ) is quantized we must have 1 π inf Rj2 h 1j n 2 and hence cG (Tquant ) 12 h. By the same argument as above, one then concludes that the projection of any quantized Lagrangian manifold Vquant lies inside a simple surface with area at least 12 h. Property (B) has the following consequence announced after the proof of (A): (C). Any admissible subset B of phase space carrying a quantized motion has symplectic capacity c(B) 1 2 h. Proof. It suffices to show that there exists a canonical transformation such that cG (F (B)) 12 h. The projection of B on any of the qj , pj planes a fortiori contains the projection of the motion. It follows from (B) that the infimum R of the radii of all cylinders Zj such that F (B) ⊂ Zj for some canonical transformation F must √ satisfy R h¯ , and hence cG (F (B)) 12 h. ✷ Discussion and possible applications The generalized quantum cells (‘blobs’) we have introduced in this Letter differ from the traditional cells used in quantum statistical mechanics: they can have arbitrary shapes and sizes. This is due to the fact that their definition does not make use of the notion of phase space volumes: the symplectic capacity of a quantum blob is
N(λ E) ∼
Vol(B) (2π h¯ )n
(6)
expressing the asymptotic value for E → ∞ of the number of quantum states with energy E in terms of the volume within the corresponding energy shell, measured in units of (2π h¯ )n . If B is approximately a ball with volume R we have Vol(B) ≈ π n R 2n /n! and hence 1 c(B) n . N(λ E) ∼ (7) n! 2π h¯ This formula shows that for almost spherical energy shell Weyl’s rule (6) can be expressed in terms of the symplectic capacity of the interior of energy shell measured in terms of quantum blobs. Now, it would be very interesting to generalize (7) to arbitrary (bounded, or unbounded) energy shells. This requires, however, new results about ‘symplectic packing’; this problem is notoriously difficult, and very little is known about it outside a few particular cases (see the review article [5] by McDuff). Another application of the notions introduced in this Letter could be the study of ‘dequantization’—i.e., the classical limit of quantum mechanics. As we have shown in [1] there exists an uncertainty principle in classical (Hamiltonian) mechanics which is formally identical to the Heisenberg inequalities. The proof of this uses the same techniques as those who lead to the linear version of Gromov’s theorem: quantum mechanics (at least in its semiclassical formulation) is much closer to classical mechanics than one is usually inclined to believe (this is also made apparent, but from a different point of view, in Hall and Reginatto’s paper [4] mentioned at the beginning of this Letter). One could thus perhaps study dequantization by showing that a classical version of the blobs introduced above exist, and exploit there properties to see exactly where the need for Planck’s constant h emerges. Finally, as suggested by one of the Referees, a very interesting application would be to investigate how the notions developed in this Letter could fit in QFT: using the fact that the quantum blob capacity of is independent on the number of degrees of freedom, one could envisage that the limit case of QFT could be
M.A. de Gosson / Physics Letters A 317 (2003) 365–369
easier to control than with the usual methods. (One could investigate, for instance, what happens when a Kadanoff-like R.G. transformation decimates the number of degrees of freedom.) We hope to come back to these fascinating and important questions in a future work.
Acknowledgements It is a great pleasure for me to thank the two referees of this Letter for extremely helpful comments and suggestions.
References [1] M. de Gosson, J. Phys. A: Math. Gen. 34 (2001) 10085.
369
[2] M. de Gosson, J. Phys. A: Math. Gen. 35 (2002) 6825. [3] M. Gromov, Invent. Math. 82 (1985) 307. [4] M.J.W. Hall, M. Reginatto, J. Phys. A: Math. Gen. 35 (2002) 3289. [5] D. McDuff, Notices Amer. Math. Soc. 45 (8) (1998) 952. [6] V.P. Maslov, Théorie des Perturbations et Méthodes Asymptotiques, Dunod, Paris, 1972 (original Russian edition: 1965). [7] V.P. Maslov, M.V. Fedoriuk, Semi-Classical Approximations in Quantum Mechanics, Reidel, Boston, 1981. [8] J. Moser, Trans. Amer. Math. Soc. 120 (1965) 286. [9] E. Polterovich, The Geometry of the Group of Symplectic Diffeomorphisms, in: Lectures in Mathematics ETH Zürich, Birkhäuser, Basel, 2001. [10] J.-C. Sikorav, Mém. Soc. Math. France (N.S.) 46 (1991) 151.
Phase space quantization and the uncertainty principle Maurice A. de Gosson a,b a Blekinge Institute of Technology, SE-371 79 Karlskrona, Sweden 1 b University of Colorado at Boulder, Boulder, CO 80302, USA
Received 10 June 2003; received in revised form 31 August 2003; accepted 7 September 2003 Communicated by J.P. Vigier
Abstract We replace the usual heuristic notion of quantum cell by that of ‘quantum blob’, which does not depend on the dimension of phase space. Quantum blobs, which are defined in terms of symplectic capacities, are canonical invariants. They allow us to prove an exact uncertainty principle for semiclassically quantized Hamiltonian systems. 2003 Elsevier B.V. All rights reserved. PACS: 03.65.Ta; 03.65.Sq; 02.40 Keywords: Quantum cells; Quantum uncertainty; Symplectic non-squeezing; Lagrangian tori
In [4] Hall and Reginatto remark that ‘. . . uncertainty relations expressed as imprecise inequalities are not enough to pin down what the essence of what is not classical in quantum mechanics’. A similar statement certainly applies to the ‘coarse graining’ of phase space into ‘quantum cells’ familiar from semiclassical and statistical mechanics. To paraphrase Hall and Reginatto, imprecise notions of quantum cells are not enough to pin down the essence of phase space quantization! In fact the usual ‘definitions’ of such cells are unprofessionally vague and go no further than Heisenberg’s 1932 discussion of the microscope; they are a heuristic device from the beginning years of quantum mechanics, and have no fundamental meaning. For instance, a very common
E-mail address: [email protected] (M.A. de Gosson). 1 Current address.
0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2003.09.008
choice is to use cubic boxes with volume hn (n the number of degrees of freedom) and whose faces lie in the conjugate planes qj , pj . This choice is thought to be consistent with Heisenberg’s uncertainty principle written ‘approximately’ as pj qj h. What about spherical cells with same volume hn ? Since the volume of a ball with radius R in phase space R2n is π n R 2n /n!, Stirling’s formula shows that for large n the area of the projection of the sphere on any plane is πR 2 ∝ πnh; this number goes to infinity with the number of degrees of freedom of the system, while the area of the projection on a conjugate plane of a cubic cells remains equal to h. Last, but not least, any definition of quantum cells directly based on the Heisenberg relations is doomed to be badly dependent on the choice of coordinates. This is because the inequalities pj qj h are not canonically invariant, except for permutations pj ↔ qj or rescalings: pj → λpj , qj → pj /λ. If, for
366
M.A. de Gosson / Physics Letters A 317 (2003) 365–369
instance, we rotate the pj , qj axes by an angle of π/4 these inequalities become pi 2 − qi 2 h in the new coordinates. What should a decent definition of a quantum cell be in this case? Our purpose is to propose a precise definition of the notion of quantum cell avoiding these drawbacks. Our definition will not depend on the number of degrees of freedom; it is hence an intensive quantity, as opposed to the usual definitions in terms of volume. The definition of these cells—which we prefer to call ‘quantum blobs’—is made possible by a careful use of two results from symplectic topology, the non-squeezing theorems of Gromov and Sikorav. The first has been used in our previous work [1] where we showed that the ground energy level for completely integrable systems can be recovered by making a simple topological assumption involving Planck’s constant. Other possible applications and extensions are discussed at the end of this Letter. Two non-squeezing results Canonical transformations are not only volume preserving mappings, but they enjoy in addition a very special property. This property, called the ‘non-squeezing theorem’, alias the ‘principle of the symplectic camel’, was proven by Gromov [3] in 1985. His theorem says that if a phase space region R (in any number of degrees of freedom) contains a ball with given radius R, then if we deform R using canonical transformations only, the area of the projection of R on any of the conjugate planes qj , pj will never decrease beyond its original value πR 2 (whereas the areas of projections on non-conjugate planes such as q1 , p2 , q1 , q2 , etc. can become arbitrarily small). Gromov’s theorem was the first of a whole constellation of new results in symplectic topology (we will describe another non-squeezing result below). It is equivalent to the existence of symplectic capacities. These are mappings associating to every subset B of phase space a number 0, or +∞, and having the following properties: (I) c is a canonical invariant: c(f (B)) = c(B) for every canonical transformation f ; (II) c is monotone: c(B) c(B ) if B ⊂ B ; (III) c is 2homogeneous: c(λB) = λ2 c(λB) for every real λ; and, finally, (IV): c B(R) = c Zj (R) = πR 2 ,
where B(R) is a phase space ball with radius R and Zj (R) any phase space cylinder with same radius and based on the qj , pj plane. The properties (I–IV) are strongly reminiscent of those of the usual area of a plane surface. One proves that in fact the only symplectic capacity in the phase plane (n = 1) is precisely area. It turns out that Gromov’s theorem is equivalent to the existence of two distinguished symplectic capacities cG and cG (the ‘lower’ and ‘upper Gromov capacities’) such that cG (B) c(B) cG (B)
(1)
for all B ⊂ R2n and every symplectic capacity c. The Gromov capacities are defined as follows: cG (B) = πR 2 where R is the supremum of the radii of all balls that can be sent in B using canonical transformations; cG (B) is the infimum of the radii of all cylinders Zj (R) into which B can be sent using canonical transformations. Another essential result is that of Sikorav [10], which is stated and proved in detail in Polterovich [9, Theorem 1.1C, p. 2, and §4.3]. Sikorav’s result is about Lagrangian tori; it says the following: let Sj1 (R) be the circle qj2 + pj2 = R 2 lying in the qj , pj plane, and T(R) = S11 (R) × · · · × Sn1 (R) the Lagrangian torus obtained by taking the Cartesian product of all these circles. Then there exists a canonical transformation sending T(R) in the cylinders Zj (r) if and only if R r; in other words: cG T(R) = πR 2 . (2) In view of the definition of the upper Gromov capacity cG Sikorav’s theorem means that the infimum of the radii of all cylinders Zj into which B can be sent using canonical transformations is equal to R, the radius of each factor Sj1 (R) of the torus T(R). Quantum blobs Let B be a subset of R2n . It follows from Gromov’s theorem and the inequalities (1) that if there exists a canonical transformation F such that B(R) ⊂ F (B) ⊂ Zj (R) then c(B) = πR 2 for every symplectic capacity c. We will call such sets B admissible. The image of an admissible set by a canonical transformation is of course still admissible.
M.A. de Gosson / Physics Letters A 317 (2003) 365–369
We will call quantum blob any admissible subset Bquant of R2n such that c(Bquant) = 12 h. In view of Gromov’s theorem the projection of Bquant on any of the conjugate coordinate planes qj , pj encloses a surface of area at least 12 h. However, to the difference of any traditional quantum cell, a quantum blob can be unbounded, and √even have infinite volume: any of the cylinders Zj ( h¯ ) is a quantum blob in its own right. Another difference is that while the volume of a cell depends on the number of particles, that is on the dimension of phase space, quantum blobs are defined in terms of the quantum of action which has the dimension of an area. We also notice that the notion of quantum blob is a canonical invariant, that is c(f (B)) = c(B) for every canonical transformation f : R2n → R2n (this follows from the very definition of a symplectic capacity). From now on we assume that H = H (q, p) is a completely integrable Hamiltonian. The semiclassical motion takes place on Lagrangian manifolds (‘invariant tori’) V which we assume quantized by the usual EBK condition 1 1 p dq − m[γ ]h¯ is an integer 0, 2π 4 γ
for every one-cycle γ is on the manifold V and m[γ ] its Maslov index (see Maslov [6]; Maslov and Fedoriuk [7]; also de Gosson [2]). We will call such motions ‘quantized motions’. Let us focus in particular on the ground state. The EBK condition is in this case 1 1 (3) p dq = m[γ ]h¯ . 2π 4 γ
We will denote a Lagrangian manifold satisfying this condition by Vground . We claim that: (A). Any motion on Vground takes places on a quantum blob (in [1] we proved the converse of this property). Proof. Passing to action-angle variables (I, φ) and defining then new canonical variables by Qj = 2Ij × cos φj and Pj = 2Ij sin φj the motion takes place on tori T = S1 2I1 (0) × · · · × Sn 2In (0) . Now, the EBK condition (3) for the ground level imposes that Ij (0) = 12 h¯ (because the Maslov index is
367
equal to 2 along circles), hence the ground-level torus is just √ √ Tground = S1 h¯ × · · · × Sn h¯ . In particular, the trajectory winds around each cylindrical quantum blob √ Zj h¯ : Q2j + Pj2 = h¯ . (4) Returning to the original coordinates q, p amounts to use successively the canonical transformations (Q, P ) → (I, φ) and (I, φ) → (q, p); this proves completely the statement since quantum blobs are canonical invariants. ✷ We will see below (property (C)) that the result (A) is actually sharp, in the sense that a quantized motion cannot be carried by an admissible set with symplectic capacity < 12 h. Quantum uncertainty We are going to precise statement (A). Let us call ‘simple surface’ in a plane a surface whose boundary can be smoothly deformed into a circle. We contend that following topological version of the Heisenberg uncertainty principle holds: (B). Any simple surface in a qj , pj plane containing the projection of a quantized motion has area at least 12 h; this condition remains true in all canonical coordinates. Proof. We begin by noting that in view of (2) we have cG (Vground) = 12 h. Let now λj be a simple closed curve in the qj , pj plane such that the surface Λj it bounds contains the projection of Vground . Let us first assume that λj is a circle Sj1 (R). Then Vground ⊂ Zj (R) and hence 1 h = cG (Vground) cG Zj (R) = πR 2 2 proving the claim in that case. In the general case choose a diffeomorphism f of the qj , pj plane taking Λj into a circle Sj1 (R) with same area; in view of a classical result of Moser and Dacorogna [8] we may assume that f is area-preserving. Let F be the phase space mapping transforming (qj , pj ) into f (qj , pj ) and leaving all other coordinates unchanged; that mapping is canonical, and the projection of F (Vground)
368
M.A. de Gosson / Physics Letters A 317 (2003) 365–369
lies inside Sj1 (R). Now cG (Vground) = cG F (Vground) cG Zj (R) = Area(Λj )
1 2 h, and this in any number of degrees of freedom. Can
we use them to count quantum states in the usual way? Consider Weyl’s rule (5)
hence again Area(Λj ) 12 h. That the result holds in all canonical coordinates follows from the invariance of the EBK conditions under canonical transformations. ✷ What about the excited states of an integrable Hamiltonian system? Sikorav’s theorem obviously implies that cG S11 (R1 ) × · · · × Sn1 (Rn ) π inf Rj2 . 1j n
It follows that if the torus S11 (R1 ) × · · · × Sn1 (Rn ) is quantized we must have 1 π inf Rj2 h 1j n 2 and hence cG (Tquant ) 12 h. By the same argument as above, one then concludes that the projection of any quantized Lagrangian manifold Vquant lies inside a simple surface with area at least 12 h. Property (B) has the following consequence announced after the proof of (A): (C). Any admissible subset B of phase space carrying a quantized motion has symplectic capacity c(B) 1 2 h. Proof. It suffices to show that there exists a canonical transformation such that cG (F (B)) 12 h. The projection of B on any of the qj , pj planes a fortiori contains the projection of the motion. It follows from (B) that the infimum R of the radii of all cylinders Zj such that F (B) ⊂ Zj for some canonical transformation F must √ satisfy R h¯ , and hence cG (F (B)) 12 h. ✷ Discussion and possible applications The generalized quantum cells (‘blobs’) we have introduced in this Letter differ from the traditional cells used in quantum statistical mechanics: they can have arbitrary shapes and sizes. This is due to the fact that their definition does not make use of the notion of phase space volumes: the symplectic capacity of a quantum blob is
N(λ E) ∼
Vol(B) (2π h¯ )n
(6)
expressing the asymptotic value for E → ∞ of the number of quantum states with energy E in terms of the volume within the corresponding energy shell, measured in units of (2π h¯ )n . If B is approximately a ball with volume R we have Vol(B) ≈ π n R 2n /n! and hence 1 c(B) n . N(λ E) ∼ (7) n! 2π h¯ This formula shows that for almost spherical energy shell Weyl’s rule (6) can be expressed in terms of the symplectic capacity of the interior of energy shell measured in terms of quantum blobs. Now, it would be very interesting to generalize (7) to arbitrary (bounded, or unbounded) energy shells. This requires, however, new results about ‘symplectic packing’; this problem is notoriously difficult, and very little is known about it outside a few particular cases (see the review article [5] by McDuff). Another application of the notions introduced in this Letter could be the study of ‘dequantization’—i.e., the classical limit of quantum mechanics. As we have shown in [1] there exists an uncertainty principle in classical (Hamiltonian) mechanics which is formally identical to the Heisenberg inequalities. The proof of this uses the same techniques as those who lead to the linear version of Gromov’s theorem: quantum mechanics (at least in its semiclassical formulation) is much closer to classical mechanics than one is usually inclined to believe (this is also made apparent, but from a different point of view, in Hall and Reginatto’s paper [4] mentioned at the beginning of this Letter). One could thus perhaps study dequantization by showing that a classical version of the blobs introduced above exist, and exploit there properties to see exactly where the need for Planck’s constant h emerges. Finally, as suggested by one of the Referees, a very interesting application would be to investigate how the notions developed in this Letter could fit in QFT: using the fact that the quantum blob capacity of is independent on the number of degrees of freedom, one could envisage that the limit case of QFT could be
M.A. de Gosson / Physics Letters A 317 (2003) 365–369
easier to control than with the usual methods. (One could investigate, for instance, what happens when a Kadanoff-like R.G. transformation decimates the number of degrees of freedom.) We hope to come back to these fascinating and important questions in a future work.
Acknowledgements It is a great pleasure for me to thank the two referees of this Letter for extremely helpful comments and suggestions.
References [1] M. de Gosson, J. Phys. A: Math. Gen. 34 (2001) 10085.
369
[2] M. de Gosson, J. Phys. A: Math. Gen. 35 (2002) 6825. [3] M. Gromov, Invent. Math. 82 (1985) 307. [4] M.J.W. Hall, M. Reginatto, J. Phys. A: Math. Gen. 35 (2002) 3289. [5] D. McDuff, Notices Amer. Math. Soc. 45 (8) (1998) 952. [6] V.P. Maslov, Théorie des Perturbations et Méthodes Asymptotiques, Dunod, Paris, 1972 (original Russian edition: 1965). [7] V.P. Maslov, M.V. Fedoriuk, Semi-Classical Approximations in Quantum Mechanics, Reidel, Boston, 1981. [8] J. Moser, Trans. Amer. Math. Soc. 120 (1965) 286. [9] E. Polterovich, The Geometry of the Group of Symplectic Diffeomorphisms, in: Lectures in Mathematics ETH Zürich, Birkhäuser, Basel, 2001. [10] J.-C. Sikorav, Mém. Soc. Math. France (N.S.) 46 (1991) 151.